Performance Rating (Linear Approximation)

A performance rating asks a narrow question: forget what you were rated going in, what were these particular results worth? It is the honest way to talk about an unrated player's first event, the right way to describe a breakout tournament, and the number a norm in international chess is defined against.

The exact definition is iterative, and this page gives an approximation. Properly, the performance rating is the rating whose expected score against this exact field would equal the score actually made. There is no closed form for it: you guess a rating, work out the expected score against each opponent separately using the Elo curve, add them up, compare with the real score, and adjust. A few passes converge. The relation on this page is a straight line fitted to that curve — the field's average rating, plus 400 points for every whole point of score above an even split — and it is the form FIDE has long published for quick work. The two agree well near an even score and drift apart toward the extremes, where the real curve flattens and the line does not.

Two failures are worth knowing about before you quote a performance rating at anyone.

A perfect score has no performance rating at all. Not a very high one — none. No finite rating has an expected score of exactly 1 against a real field, so the equation being inverted has no solution, and the same applies to a zero score at the other end. Every convention for handling it is a patch: add 400 to the field average, cap the answer, discard the event, or treat the perfect score as one point less. They give different answers and none of them is right, because the question has no answer.

And averaging the opponents throws away information. Two events with the same field average are treated identically, whether every opponent sat within fifty points of that average or the field was a mixture of grandmasters and beginners. The exact iterative definition does not have this problem, because it evaluates each opponent separately — and the more spread out the field, the further apart the two methods land.

The deeper caution is about sample size. A performance rating from nine games has an enormous standard error — on the order of a hundred points, which is more than most of the differences people use it to argue about. A single tournament is a small sample, a spectacular result is often mostly luck, and one performance rating well above a player's established rating is very weak evidence that anything has changed. That is not a defect of the formula; it is a fact about nine games.